More on approximate Ramsey properties

Using the metric version of the KPT correspondence, we prove that the automorphisms groups of several limits of finite dimensional operator spaces and systems are extremely amenable, including the Gurarij space and its non-commutative version $\mathbb{NG}$. Dually, we prove that the universal minimal flow of the Poulsen simplex $\mathbb P$ is $\mathbb P$ itself, and again similarly for its non-commutative version $\mathbb{NP}$. The approximate Ramsey properties (ARP) we find are consequence of the dual Ramsey Theorem (DRT) by Graham and Rothschild. In a similar way, we will see present an approximate Ramsey property for quasi-equipartitions and how to use it to deduce the ARP of the family $\{\ell_p^n\}_n$, $1\le p\neq 2<\infty$. We will also discuss the reformulation of the (ARP) of $\{\ell_p^n\}_{n\in \N}$ as a weak version of a multidimensional Borsuk-Ulam theorem. Finally, we will see that - the DRT is a particular case of a factorization theorem for 0-1 valued matrices, - the Graham-Leeb-Rothschild Theorem on grassmannians over a finite field $\mathbb F$ is a particular case of a factorization theorem for matrices with values in $\mathbb F$, and - the ARP of $\{\ell_p^n\}_n$ is a particular case of a factorization theorem for matrices with values in $\mathbb R$ or $\mathbb C$. This is a joint work with D. Bartosova, M. Lupini and B. Mbombo, and with V. Ferenczi, B. Mbombo and S. Todorcevic.Non UBCUnreviewedAuthor affiliation: Universite Paris 7Facult